Started hand with $110 (~36BB)

Under the gun with JJ raised to $12 (4BB)

2 Callers, Button has about $300 behind and raises to $29 (~9BB)

I call and two other players call as well

Flop: A74

Checks around to button, he bets $100.

Hero? Thoughts on earlier action?

Welcome to the forum, OP. Please don't post results in the future, as it biases people's advice. I removed them for you.

This game sounds pretty bad, with a 40BB max bi? How is the rake? Unless it is very good, this might be a nigh unbeatable game long-term.

Raise pre seems fine. Maybe could be a touch bigger, depending on what is normal for the game.

Calling the 3-bet seems iffy this short. You'll be OOP, will have already put in a quarter of your stack, an overcard usually comes, and your stack to pot ratio is a measly 1-1 or lower if others call too. If you were deeper, you could call to set mine. This short, it's shove or fold, imo. Since most players don't get out of line with three-bets at this limit, I'd fold it unless V had a maniac image.

AP, checking flop is fine, and I'd generally be check/folding. Someone has at least an ace, unless the flop checks through. In fact, the only value hand in a normal V's three-bet range that you beat pre-flop (AK), is now ahead. Never calling.

All that said, going forward, unless this is a wild game that builds 100BB stacks quickly and has a decent rake, I think the right play is not to buy in at all.